Dixon elliptic functions

In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map from each regular hexagon in a hexagonal tiling to the whole complex plane. Because these functions satisfy the identity ${\displaystyle \operatorname {cm} ^{3}z+\operatorname {sm} ^{3}z=1}$, as real functions they parametrize the cubic Fermat curve ${\displaystyle x^{3}+y^{3}=1}$, just as the trigonometric functions sine and cosine parametrize the unit circle ${\displaystyle x^{2}+y^{2}=1}$.

They were named sm and cm by Alfred Dixon in 1890, by analogy to the trigonometric functions sine and cosine and the Jacobi elliptic functions sn and cn; Göran Dillner described them earlier in 1873.